"The Hadamard product and the free convolutions" by Arijit Chakrabarty
 

The Hadamard product and the free convolutions

Article Type

Research Article

Publication Title

Statistics and Probability Letters

Abstract

It is shown that if a probability measure ν is supported on a closed subset of (0,∞), that is, its support is bounded away from zero, then the free multiplicative convolution of ν and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained.

First Page

150

Last Page

157

DOI

10.1016/j.spl.2017.04.009

Publication Date

8-1-2017

Comments

Open Access, Green

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